The Real And Complex Number Systems

Since f a f p 0, we have f q 1 0whereq 1 a, p and since
f p f b 0, we have f q 2 0whereq 2 p, b by Rolle’s Theorem. Since
f q 1 f q 2 0, we have f c 0wherec q 1 , q 2 by Rolle’s Theorem.
5.22 Assume f has a finite derivative in some interval a, .
(a) If fx 1andf x c as x , prove that c 0.
Proof: Consider fx 1 fx f y where y x, x 1 by Mean Value Theorem,
since
lim fx 1
x
which implies that
limfx 1 fx 0
x
which implies that (x y )
lim
x f y 0 y
lim fy
Since f x c as x , we know that c 0.
Remark: (i) There is a similar exercise; we write it as follows. If fx L and
f x c as x , prove that c 0.
Proof: By the same method metioned in (a), we complete it.
(ii) The exercise tells that the function is smooth; its first derivative is smooth too.
(b) If f x 1asx , prove that fx/x 1asx .
Proof: Given 0, we want to find M 0 such that as x M
fx
x 1 .
Since f x 1asx , thengiven 3 ,thereisM 0 such that as x M ,we
have
|f x 1|
3 |f x| 1
3
By Taylor Theorem with Remainder Term,
fx fM f x M
fx x fM f 1x f M ,
then for x M ,
fx
x 1 fM
x |f 1| f M
x
fM
x 3 1 3
Choose M 0 such that as x M M ,wehave
fM
x
3 M
and x
M
x by (*)
*
**
/3
1 3 . ***
Combine (**) with (***), we have proved that given 0, there is a M 0 such that as
x M, wehave
fx
x 1 .
That is, lim x
fx
x 1.